Quaternion Quantum Mechanics

Transcript

hello my name is chantal this is a presentation about quaternion quantum mechanics this is actually a um presentation based on marek dominiovsky's talk he recently gave a seminar on his most recent paper called foundations of the quaternion quantum mechanics it's a paper published in entropy in 2020 and i think it was really an important presentation and i thought i would like to try to record it so in order to make it more accessible to other people before i can actually get into the details of this presentation i want to first give some introduction on the topic and also some motivation why you wouldn't even talk about this at all and my goal is to make this sort of difficult topic really understandable so i will go slowly and i'll explain everything in detail and feel free to skip obviously if it's too easy for you so first i'll give the overview um interpretations of quantum mechanics and about phonons then i'll talk about the haagen-kleiner plank crystal which was developed by both um hugging leonard and mark danielewski we'll talk a little bit about the motivation what this could mean then i'll briefly introduce you to quaternions because that's not something many people use every day because the whole idea here is that we'll build up quantum mechanics using quaternions and we'll see why and after that we'll get into marek's presentation so many slides are taken from marek with his permission and i highlighted whenever i remembered on each page which uh slides were taken from marek's presentation and of course there will be a lot of um links in the description and also at the end of the talk so you all know there are many interpretations of quantum mechanics right starting from the copenhagen interpretation which is basically um developed by born heisenberg which is sort of mainstream right now the idea here is that the wave function is not real and that there's this wave function collapse upon a measurement which is immediate then there's the broly boom um interpretation which is also referred to as a pilot wave theory here the wave function is actually real it's more it's like a guiding it's guiding the equations it's not local um it's deterministic but there's still this notion of particle involved so it's only this pilot wave that guides the particle then there's a statistical interpretation which is really a minimalist ensemble interpretation the wave function is really more like an abstract quantity there are few assumptions probably the fewest of all then i'm sure you all know the many worlds interpretation everett the idea he here is that you know there's no wave function collapse instead the universe splits into different universe parallel universe each time a measurement happens of course one challenge here is that you know it kind of violates physical laws right but still you know it's valid interpretation then there's super determinism and there are many more interpretations i only listed a few so i'm sorry if your favorite one is not here but there's also i would i've seen this word called neoclassical which is really a realist interpretation saying that there's something some reality underneath similar to the broly bomb interpretation except they're in this model here they're only waves so why would we even talk about this right well clearly you can see there are many possibilities so given that there are so many possibilities people have there's no agreement so it is still valid to talk about this and um in fact um murray gilman he had he got a nobel prize in about development of particle physics he said niels bohr brainwashed a whole generation of theorists into thinking that the job of finding an interpretation of quantum mechanics was done 50 years ago which is clearly not true as you as you can see right um and also schroedinger shooting at himself in 1952 in dublin he said uh let me say at the outset that in this discourse i am opposing not a few special statements of quantum physics health today i'm opposing as it were the whole of it i am opposing its basic views that have been shaped 25 years ago when max born put forward his probability interpretation which was accepted by almost everybody so he was actually really upset because he believed there is a real wave not just the probability so this is the idea that this is also probability is an interpretation now what would be a classical v what would this wave be what how could we imagine this being a real wave so let's first look at something actually classical i mean classical i mean an actual phenomenon we can see in crystals and this is something called phonons you may have heard of it sometimes it's referred to as particle of sound or particle of heat it's something that's used in condensed matter physics and this again this is totally classical so we are talking about a real real crystal that vibrates okay and these vibrational modes as you know this crystal vibrates it you will see sometimes these quantized motions these quantize it they refer to this as quasi particles so it these these vibrational modes they behave similar to particles right they move around they behave like bosons um they're basically quantized sound waves the difference here is that here we know it is just a wave because obviously we know sound is a wave so and this is in a solid right so there's no question about even that this is a particle we all know this is just a vibration but it kind of looks like a particle and so it's also called a a quasi-particle we and actually it has a lot of properties to photons so you know both photon and phonon are bosons you know photon is referred to as a particle of light photon is a particle of sound both are quantized energies both have this wave particle duality both actually their energy is h times f and of course there's differences too right right photons are electromagnetic waves phonons or excitations of a group of atoms but then makes you kind of wonder right i mean if a phonon which is clearly a wave it has no particles there's no particle associated with this why shouldn't this be possible for photons as well right so the reasoning that oh we need a particle is maybe not true who knows and interesting you can even do a double slit experiment with phonons we all know how it works with photons right this picture on the right is a experiment using polarity on surface polaritons there is a movie somewhere but i couldn't find it anymore so you can actually see the wave pattern generated by this young double slit experiment and interestingly right while for photons people always say oh this is kind of weird you know because when you measure it suddenly it's in one place well for phonons there's no it's the same thing it's the same math it's the same experiment but clearly it's just a wave right there's nothing weird about it it's just a wave and so um i makes me wonder why should it be so different for photons right if the math is the same as for phonons so and there's actually a paper oh there are several paper on phonons one of them is mentioned here and it says the quantum mechanical properties of phonons in a one-dimensional lattice are studied with the conclusion that the phonon behaves in all essential respects as a normal quantum particle wave function collapse of the phonon state is shown to occur in an automatic way when an observation is made this gives possible insight concerning the nature of wave function collapse in the general particle case so when you think about this for phonons we know these are just vibrations and the measurement basically transforms this wave into another state it's a dynamic process that takes time so you know couldn't it be the same for photons it's just a question right so to me that's one of the motivations to look further into this topic to see if that's possible or not let's briefly talk about being cough speech space time income 60 um you all know that you know this is composed of three spatial and one time dimension and gravity is caused by space-time curvature so we picture this sort of it's kind of hard to picture because it's four-dimensional right but there's this curvature this abstract curvature in space time that causes gravity you probably didn't know but there is an equivalent mathematically equivalent model initially developed by han kleinert and basically what he did he did a simple coordinate transformation so instead of space time we have space density everything else stays the same and this model you can imagine a 3d coordinate system with changes in density right so wherever there is a lot of matter the grid is a little bit condensed so it curves this space and gravity in this model is actually refraction it's an optical mechanical analog to general relativity now you think oh that's crazy right but the people in condensed metaphysics they use optical mechanical analogy to general relativity they use the metric tensor code to calculate things in condensed matter and um marek danielewski he has also written about this and his work that i will present later in this talk is actually based on this planck clander okay this is hot and cleaner and someone created a portrait of this crystal obviously it would probably not look like this so this is now a slide taken from a marathon levski's talk a visualization of of course this is this is just a visualization but it maybe gives you some intuition of what this could be and let's play this so basically this is like image like a 3d you know elastic solid okay with that can have deformation so you can imagine it can have transverse waves transfer faith longitudinal waves you know it can have basically compression and torsion inside of it but the crystal as a whole stays it doesn't break it stays in one piece it doesn't change overall in volume this is the whole idea of this planck planet crystal now you think oh this is far-fetched right but you know refraction is every if you wear glasses you know you know what refraction is and you can actually do an experiment in your kitchen um and i got this from youtube i will put the link in the description as well so what you can do you can create a gradient of water and sugar so sugar gradient so you have lots of sugar at the bottom unless you're on top and when you shine a laser through it because of the gradient so that's graded from top to bottom okay um when you shine a laser through it the laser light will start to bend that's simply caused by refraction because there's a gradient right which changes the speed of light actually there's also um another analogy that's optical black holes there are multiple papers on this this one is from nature originally based from nature but there was a there's a link also at the end of the presentation to this here is from the experiment and on the right side is the theory so what they did they had a sort of a meta material a optical material which has an increased also gradient um an increasing refractive index and when they shine light towards it the light would bend because of the refraction because if it's a gradient at some point you know if the refractive index is large enough the light can no longer escape so it's very similar to a black hole except here it's an optical black hole i think that's very cool there are also black holes of sound so so maybe this is not all that mysterious and maybe this optical analogous gravity is maybe not so strange so what kind of properties would this crystal have um the length of this grid from one one element to the next is called the planck length lp it's in the order of 1.1 to the 10 to the minus 35 meters so very very small the planck mass is this and there's also the um planck time here is the frequency 1.8 times 10 to the 43. so that's basically the fastest process that's possible and this would be the smallest thing possible and these values are actually from the national institute of standards and technology nist so in summary what is this planck clannad crystal you can view it you know you i'm sure you have heard of term fabric of space-time right when we talk about minkowski space-time so there is this notion that the vacuum is something like well it can bend right it's like the the grit now in this case it is the fabric of space-time but it's space density so you have three coordinates of space and one of density um you know it's it's you can more picture it like a like a grid like a 3d grid it is an elastic solid which can have compression and torsion and the analog of gravity is an optical analog and even though you may think this is all strange even maxwell talked about this so he he actually said in 1869 the assumption therefore that gravitation arises from the action of the surrounding medium leads to the conclusion that every part of this medium possesses one undisturbed and enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction i'm unable to understand in what way a medium can possess such properties i cannot go any further in this direction in searching for the cause of gravitation so he thought that there must be some kind of medium and that matter would influence it so basically when there's a lot of matter like an earth or sun that medium would be more dense right which which increases the refractive index which causes um gravity and actually even einstein was not you know had these thoughts so this as you can see this is not just some crazy idea um you know famous a lot of famous people have these thoughts he um said at the reich's university in leiden he said more careful reflection teaches us however that the special theory of relativity does not compel us to deny ether we may assume the existence of an ether only we must give up ascribing a definite state of motion to it recapitulating we may say that according to the general theory of relativity space is endowed with physical qualities in this sense therefore there exists an ether according to the general theory of relativity space without ether is unthinkable for in such space there not only would be no propagation of light but also no possibility of existence for standards of space and time measuring rods and clocks nor are therefore any space-time intervals in a physical sense so even einstein said there must be something like a fabric of space-time right so even though a lot of um concepts of ether have been disproven so we know there's for instance no gases no gas or something you know there's no ether wind there have been a lot of experiments but but there's has but a real a solid ether has not been disproven for instance an elastic solid is actually possible why a solid because well it must support transverse waves it must support different wave shapes right so that's actually still possible so this is just to give some motivation why should we even think about this and also maybe that's not so crazy now before we can actually dive deeper i'll have to i want to give you a brief overview of quaternions because i mean it was kind of new to me so and it took me a while to understand it so i think it's not all of you might know about about quaternions so maybe you have heard about the gimbal lock if you fly airplanes or if you even if you play computer games maybe you have heard of this um basically when you have a gyroscope rotation is measured um with three axes normally right they're orthogonal to each other but what can happen though when you fly say a plane or actually it happened in apollo 11 when you turn 90 degrees like in this case the airplane moves upwards so that the green axis is now aligned with the blue axis this is called a gimbal lock because now these both axes are combined and you can no longer separate them and so when you have a gyroscope for instance and you measure your your um orientation you know the system can no longer tell what orientation you have that's really they even had a special button in the apollo 11 that's what happened they even have a special warning light for gimbal lock and what this happened they had to manually start to navigate because the system wouldn't work anymore so this is just one example of the problems we have with dealing with rotations in space and why maybe a different system might be better than using the standard euler angles now if you have been working in computer graphics or if you are a game developer or you just play around in unity you will have heard about quaternions so what are they they are described with four numbers one real and one scalar and three complex complex number now don't get scared by the word complex because basically all it means that three of the numbers so the last three are used to determine the axis of orientation using x y and z just like a vector similarly it you know points in the direction of the rotation axis and the first number is used to tell how much to rotate so why would you use this well imagine you have a camera in a computer game right you move your camera to another location now when you move a camera it's not just a a translation right most of the time you want to also rotate the camera around an object for instance and this movement is calculated using quaternions quaternions are very very well suited to compute smooth rotations because when you just use the oil angles you get this problem with the suddenly you know uh you get flips in and you get weird behavior and with quaternions you don't it's very easy with quaternions um there are many ways to represent them uh and in this presentation we use several of them so sometimes you just list the four numbers so again the last three are the rotation axis in in computer graphics the first one is the amount of rotation sometimes you will see i j and k where each one is a different axis like the x y and z axis sometimes you will see a key with a hat on it and sometimes you'll see the vector because the the these last three are basically vectorial and the first one is a scalar and there are some really good um youtube videos and also a there's an interactive um page where you can play with quaternions to really understand them and the link is here in the description also on the last slide so who actually invented them it was sir william rowan hamilton in 1843 the history goes that he was walking well he has been working um with the 3d 3d operations um but he he's been using vectors and he got stuck with it for years because he just couldn't figure out how to properly get this to work you know he did he thought maybe he will need um you know instead of just the i also a j basically a second complex number but it just didn't work and he was walking across this bridge in 1843 with his wife and suddenly he had the idea oh we need a third one we need a third complex number oh my god he didn't think of it before and he was so worried he might die of a heart attack right then he carved this into the bridge it's actually still so they yeah in the bridge right there and so one of the insights he had is that i squared you know for complex numbers you know that i squared is minus one and for quaternions it's similar except we have now instead of one chord axis of rotation we have three so j squared is 1 and 1 minus 1 k squared is minus 1 and i times j times k minus 1. now that sounds weird i know but it's basically if you think about this it's just diff three different coordinates um directions of course and there are many talks about these are presentations i won't have time to go into the details the people who can do this much better than i can um to ex to understand what these are interestingly um vectors are not suited at all at all for working within 3d space it sounds strange because we are all used to vectors right but in fact the problem is with vectors division is not defined did you ever learn division in school for vectors no because there's no division um there are only four division algebras literally it can be proven they're only four so we can have division you know with real numbers obviously complex numbers quaternions and octonions so you know for instance in vector vectors you have the dot and the cross product it's kind of clumsy no division and so actually quaternions make a lot of things easier in in 3d space and maxwell again he also he noticed oh my god this is really cool he said the invention of the calculus of quaternions is a step towards the knowledge of quantities related to space which can only be compared for its importance with the invention of triple coordinates by decart the ideas of this calculus as distinguished from its operations and symbols are fitted to be the greatest use in all parts of science he actually tried to reformulate electromagnetism with with quaternions but he unfortunately failed because it's not super trivial uh and others failed too unfortunately so people unfortunately they kind of gave up on it for a long time and they just used vectors because it's i guess it's easier and so unfortunately all of quantum mechanics has been built up with vectors and matrices instead of quaternions so this is where barrack comes to the rescue so of course other people have tried this too and there are other papers on this but he has now derived a lot of quarter mechanics based on quaternions and i will start to try to explain now how if i can so let's go back to the crystal okay what kind of things can happen to a crystal well it's we could call it deformation fields so you can have compression that's just denser more than less than compression to divergence which is um irrotational or you can have curl or twist which is basically the rotational the twist part right and maybe you can work i can see where this is going because um we can split up a vector field into these two parts in fact this is called a helmholtz decomposition and he has proven that any such field you can decompose into the rotational part divergence and the rotational part now based purely based on this definition it's clear that the rotation the twist of um the compression part obviously must be zero right and also the opposite the divergence of the rotational part must be zero because that's how we define it we actually split it up into these two parts and so if you look at these symbols these will come up in many places in this presentation so um you will see uh u0 as for the compression or sigma zero will be used for compression and then you have the victorian part um the uv or with the arrow or um with the youth yeah will will be denoted as the the curl so how do you actually work with deformations in an actual solid not not just this hypothetical plane crystal but in a real solid so you have a chunk of you know crystal on your table or elastic solid how do you work with this in gauche has been div he has developed the displacement mechanics how you calculate things um how mu you know if deformations in this in this cross in an elastic solid basically you have translation and rotation the point is the overall shape or size is the same so your overall you know cube is the same in volume the whole thing does not just get bigger or smaller it stays the same but internally you can have compression and twists okay um so if you can if you imagine you have in your room a big coordinate system with with rubbers rubber bands for instance in all the three directions right and you you hold it if your hand in there and you do something with it basically it means you're not going to break the coordinates the coordinate system stays you can you can change it translate you can rotate things around you can combine all these movements but you don't break it and basically he has developed a very simple formula of how how you know the equation of motion and by motion it means each individual grid element basically in there so this is um the acceleration is a combination of the gradient of the compression and the twist of the twist okay it's actually quite simple this is how it's spelled out and again this all these most of these slides are from marek danielewski's talk okay these are not mine these are from marek's talk so this is how what you would do to to model the um deformation of of an elastics of a real elastic solid and now the thing is that modeling such a thing it's not vectors are really not suited as i mentioned before um you know you cannot reduce this problem to vectors there's there's no division operation and so he so basically after trying to work with this forum like this one i mentioned that hamilton invented the use of quaternions because it makes things a lot easier the point is this there's basically a deformation field sigma so the whole thing is called sigma the whole deformation field such that one can represent the solenoidal the vector so that's the rotational part and the scalar field which is the compression part is a superposition of real and imaginary field parts at each point it's just same thing as the helmholtz decomposition right it's really the same thing all we're saying is we can build it up from compression and torsion that's it these are the symbols used so you will see them again it kind of helps to get familiar with these because otherwise it gets kind of confusing if you don't know them so um to the sigma this is the whole comp the whole deformation field is composed of the irrotational part the scalar and the vectorial part the curl right and this is the a complex conjugate is basically a minus instead of a plus and these these numbers so this this is basically a quaternion right because you need you need four numbers to represent this you need the scalar and three vector vector parts right so these are quaternions and this is how um sometimes this is written right okay so and again this is really nothing that complicated all it means you know we have the first scalar is compression and the other three the complex parts the i phi one the j if i two the k five three is is the rotation is the rotation twist in 3d space that's all it is oh i had a macro on so now when you think about this how what kind of numbers do we have in um quantum mechanics well in quantum mechanics we have something a little bit similar although not exactly we also have um compression and we have one imaginary number right if you know schrodinger's equation the i in there which bothered me all the time because i never really saw a clear explanation of why it should be there it's basically just one axis of rotation instead of three so when you think about this all it is really it's a you can project a quaternion with three uh imaginary axis with three axis of rotation to one axis it's the same thing except here you only have the i left in the in standard quantum mechanics and with quaternions you have three axes that's it but it it's totally i mean it seems to me obvious that this is compatible because you can always convert a quaternion back to a complex number by just you know setting the other by projecting the axis the 3dx down to a to a flat to the 2d space right but we'll talk a lot more about this so don't worry about this so now all the trick all the things you have to do now is uh combining gauche and helmholtz and again these slides are all from marek anilevski talks these are not my slides these i i took his slides and they combined them with mine but basically this is not my material this is morex materials just to be clear so he had the idea to combine gauging and helmholtz it's actually quite simple so remember this is the gauche equation of motion and we know we can separate out the compression and torsion part now all we have to do is apply it right it's actually quite straightforward so we do two steps first we apply the the divergence and afterwards we'll do the same thing we'll apply um the the curl the rotation so when we apply the divergence to this because of the way this was defined right we did how we split it up we know the divergence of the vectorial part must be zero because that's the rotation so their versions of the rotation must be zero so this uh turns to zero and um so that simplifies when you when you um this is turns to zero and and now all we need to do now we make a little simplification we just replace um divergence of the scalar part with sigma zero okay this is the equation you get and this is actually a longitudinal wave in three dimensions so by this simple applying helmholtz in gauchy we end up with a longitudinal wave in 3d that's nice right i mean that was pretty simple so now let's do the same thing but apply uh rotation on the you're basically using that the helmholtz decomposition we apply the rotation to gauche and here of course we know that um the scalar part so the compression part in the helmholtz decomposition has no rotation so that must be zero right so when we apply this we we apply the rotation all the red parts end up being zero we just purely paste from definition so this simplifies quite a bit it's still a bit long but um we'll go to the next page to see uh and of course please read the paper the paper has a lot more detail and some of you know i i don't want to go through all the detailed steps it will take too long but the principle is quite simple right um now what we also know is we know that the rotation of the gradient of something is zero as well and finally we make a substitution we replace all the root u the factorial part um with this symbol here okay so when you apply both it gets already simpler see because all the see that old red part goes away it's zero okay this one is replaced by um by fee and this is the same as from the previous page and now we do one final replacement we change this we simplify this with the laplacian operator and what you end up with here this is nothing else but a transverse wave right so that's pretty nice so all we did is we applied the helmholtz decomposition to gauche and we end up with longitudinal and transverse waves so both again this is just a summary again of both but of course remember we split it apart in fact we have to look at the whole thing together right we are talking about this whole deformation field now so it's a sum we have to look at both together that so it's not just either transverse wave or longitudinal it's actually combination of both right it's a combination of these waves which means we can generate all kinds of shapes of of waves any combination of transverse and longitudinal you will you will get all kinds of shapes imaginable right for instance here's one example it's just one example there's a mathematical trick you can do so um remember this is the original equation we just started with which is the combination of the transverse and longitudinal and we can we can always add something and then subtract it at the same time right it's still the same equation and we do when we do this trick we end up with the klein gordon equation which is a relativistic wave equation and it's lauren's invariant so what kind of physical constants could we actually calculate from this model can we actually calculate something from this well we have the formula now right and um what we can now do since we have the equations we cannot put in the values and see what comes out and when we do this substitution here and again details are in the paper so lp lp is the uh planck length for instance then we this can be expressed as a function of the local mass density and you can already see the g in here okay now when we do this based on the work by merrick that inevsky what we can do we can calculate g the gravitational constant and it turns out to be 6.67408 something 10 to the minus 11. and the official number i think it's from 2006 is six point six seven four two eight something ten times ten to the minus so it's very close and i think that's really amazing right we have just you know start with a simple model uh of this crystal used you know gaucho's equation of motion in helmholtz decomposition and now we end up with these constants which match actually what's been measured so i think this has to be taken seriously right this is pretty amazing i mean to me that's very amazing so and actually marek danielewski has calculated many more constants using this model this is just there are several papers you should really check out his research page he has a altogether about 190 papers and several are on on this um elastic solids and and you know in condensed matter physics but also on this clank planet crystal for instance um he has also calculated um the planck constant so he comes up he comes up with 6.626 10 to the minus 34 and the official nist number value is 6.626 it's almost the same can you see that how accurate it is this is just amazing i mean how when if you have a model that makes such predictions i think that's pretty that's really convincing to me he also calculated the speed of light using um the constants right using the young modulus in all these and he comes up again to a value that's very close to the to the official value so well we we're not stopping here so we just started right we we have derived some wave equations but we don't want to get to schrodinger right now this gets a little bit complicated and i'm not going to go into extremely detailed extreme details so please read the paper the in the in the paper there it's much more of the many more steps and there's also independence there's a lot more explanation there about two or three pages of math and i i don't want to go through this whole thing it's easier if you just read it i will show you just a few things okay because i want a more you know convey the concepts and ideas than the actual little detail steps so to derive the schrodinger equation we'll have to think about what it means to have um the gradient of a quaternion we need to define that obviously right what's the derivative what's the gradient of the quaternion and so this is called the gauche riemann operator it's represented by d it acts on paternium valid functions so functions with quaternions so d applied to this is this the gradient obviously is it's very simple actually i mean the first part remember is compression the second one is the rotational part so it's just the gradient of the compression of the the scalar part and the the curl of the of the vectorial part and that's the quaternion that's all it is the definition of quaternion so when you apply this twice you get the laplacian operator and so d corresponds physically to the gradient uh in 3d space it's not that bad not that complicated so to get to schroedinger we have to talk about energy how do we define the energy of a displacement field right yeah imagine you have this crystal and you know as it vibrates you have this motion motion of the grid elements or little particles that vibrate so there's energy in there in a movement of this grid so to define the energy per mass unit in the deformation field it's a composition of the the so this is the um the velocity of the of the of the elements in integrate of the formation field and um then we have also and of course to calculate the energy of a volume we have to do the integration right so the top one this one is the energy per mass unit and this one is the integration of a certain volume and of course we could we could also have a external field that worked that is applied to that information field so how do we get to schrodinger from this and of course again i mentioned that several steps and i won't go through all the details there are a few tricks um basically the rescaled velocity so that the velocity divided by c this the normalized velocity is related to the normalized gradient of the mechanical potential so lp is the planck length when you do when you note so you have to keep this in mind and then we are starting to we do some substitutions we um we define phi as this expression here and what's happening from here to here basically instead of um sigma and so on we replay we you put this expression in here don't you know place it again in the paper it's much more detailed then also what we do we substitute so we divide by c and of course if you divide by c we have to put it here um then we replace this this normalized velocity with the this expression here with the mechanical um what's it called the normalized gradient of mechanical potential to get this expression and um this has to be minimized and there's something called the raymond lemma which i'm not familiar with in detail so but when you do this and when you read it in the paper to step by steps then you end up with this and you can already kind of see the resembles um of the schrodinger equation and all you have to do now is basically replace some of these constants and then you end up with the time invariant schrodinger equation okay all based on quaternions now and so what does it mean instead of the i that we normally have right so the traditional schrodinger equation has the complex i in there instead of that we have i j k and now it's kind of clear what this is right this is just the rotation axis it has to do with the rotation that three makes a lot more sense to me than just i i always this always bothered me and so this to me makes a lot more sense um so also the psi of course is not just complex it's a quaternion right so so now again let's go back to this picture maybe you can see that this is really not so much different from the basically original quantum mechanics except instead of one complex number i we have three now using quaternions so i think and again when you project the quaternion onto a plane basically you you're left with the eye so it's not that much different and to me quaternion makes a lot more sense because it's much more general it's much more general and i'm looking forward to future work because what's next of course is well how do you represent spin what about the drock equation and if you know about this you know the spy spinners and the poly matrices um clearly they have to do something with with rotation so it's kind of obvious that it should be much more elegant to to formulate these things with quaternions then suddenly having these matrices inside your equation right makes no sense to me so yeah so this is a question for me i mean this is just the beginning he you know this is a paper that was just published but that's not the end of it this is the beginning so i would encourage any of you if you have any suggestions what else in what direction the research goes write in the comments and um you know uh send me an email if you like because or or do research on your own this is this is an exciting new opportunity to develop these things these techniques and you know you know what about visual visualizing all this what does spin look like and this is another from a youtube video a nice video that it's obviously not exactly like in the crystal but it would be similar because as you can see if you look at the this is called the belt trick in a way this rotation here you know it keeps rotating you see how the cube keeps rotating rotating without breaking the coordinate system right it doesn't break so you we could imagine we could visualize this in the crystal as well it doesn't actually break the the coordinate system so i imagine spin would could look similar i don't know this is just by my hypothesis so now back to the interpretations of quantum mechanics remember i had this little box here is that neoclassical so basically in this model the wave function is real it's ontological but then there are no true particles so everything we you know electrons protons neutrons photons everything exists in our world is just waves it's just different wave shapes different wave forms and anything that to us looks like a particle is just like a phonon like it it's basically quasi-particle of course it was it looks like particle because it's you know it's quantized but um there are no true particles in that sense and of course obvious question that he might have had right from the start is well you know this this sounds like there's an absolute frame of reference horrors right how could well this is clearly not compatible to special relativity if there's you know we all know there's no absolute flavor of reference well if you think about it let's just think about this for a minute okay so the question is is this lawrence invariant can't you just tell we're you know there's an absolute frame of reference can we just determine our speed if if there is this crystal can we just measure how fast we are going along uh you know the crystal and also the speed of light well it would be constant in an absolute sense meaning it's constant in the crystal so wouldn't a moving observer see a different speed of light slower well actually no because remember we are all composed of waves which means our clocks are composed of waves okay here is an example a very simple clock imagine it's a circular standing wave let's say transverse wave okay that goes in a circle and each time it goes around a circle it's one click so it goes tick tick tick right let's imagine you have two clocks the stationary one and the moving one on the right this one is moving downwards remember the speed of light is constant in the crystal okay so when the clock moves downwards you get a helical path it's no longer circular it means it's a longer path okay so in the same amount of time they travel the same distance because the speed of light is constant but so when the first clock is doing a whole revolution the second moving clock maybe only gets to here because it has to go a longer path right what does that mean it means the second clock is ticking more slowly than the first one and it's not a conspiracy you can do this on a piece of paper it's very simple actually and if if you like i'm gonna go one step further i have a simple analogy it's really my goal was to explain this to my dad so i wanted to make a simple explanation of special relativity so obviously this is not a model you can take one to one but let me do this it's it's so simple that i think you if you understand this you will totally understand special relativity okay something you can do actually on a boat so imagine you have a boat and um you have this water clock this wave clock which all it does is you have um you know a stick you or something you point in the water which generates this wave and when the wave reaches the bell it rings that's one tick off the clock okay so there's you can see the wave front how how it spreads and after some time it hits the bell and it rings the ball very simple right so that's our clock of course the model breaks down at some point because you have to look at the water speed and the wave speed and the size of the wave but this is this is detailed it's just a big concept so now let's compare a stationary boat where the wave basically travels to the club to the bell like this to a moving boat now when you have a moving boat when we you know start the clock the tick here but then the move both moves to the right what happens is the path is getting longer until the wave fronts reach the bell so and because the speed of the wave just like the speed of light is constant assuming there's it's not a river or anything um so this moving clock will take more slowly than the stationary clock that's all there is to it special that this is really old it's really not that not that difficult so now we can just draw this and all you need to know is pythagorean um thing um so basically you have the the moving boat so the boat is moving at a velocity v and all there is is this triangle and you get the lawrence factor out of this very easily you can do this on a piece of paper in like five minutes no two minutes actually so now of course the difference is in the in the crystal world we are also composed of waves right and of course in this model um the boat was not a wave he was actually uh material yeah so so but so we have to we have to use actually a crystal world to do this properly imagine there was a a phonon world so a crystal with phone on people everything composed of waves it's the same thing right so we have photon clock where um this sort of boat composed of waves as well emits a signal and it gets measured on the other side right so uh basically like a solid wave and now it's just it's the same thing you know because if the speed of light is constant in this absolute crystal or grid then the moving clock right the path is longer to reach that target here to to uh cause a click it's gotta be slower it cannot kick as click as fast because the path is long it's very obvious i mean when you think about this it's very super super obvious and again it's just a triangle the pythagorean i can't say the protagoras it's very easy calculation so special relativity is just an emergent phenomenon all wave systems have special relativity it's nothing special really here's the calculation it's really really simple so here's the links i will also put them um down in the description and if you have any other any questions i like to explain things in simple terms so if you ever wondered about anything in physics let me know maybe i'll i'll do a video on this thank you for listening and your patience